Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(10,11)(12,16)(17,18)(19,22)(20,24)(21,23)(25,26)(27,32)(28,31)(29,30);(3,7)(6,8)(9,10,13,11)(12,15,16,14)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,32)(30,31);(2,5)(6,8)(12,16)(14,15)(18,23)(19,24)(26,29)(28,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,28,30,32)(26,31,29,27);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,20)(18,19)(21,22)(23,24)(25,29,30,26)(27,32,31,28)Polycyclic presentation:
2^9|b^2=hi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5389603);